It's clear that many people who play the games and delve into the lore are aware of the Thalmor's intentions with Adamantine Tower, White-Gold, and all the others that I'm not enough of a lore-buff to know by heart. But how many, if any, people in the Empire are privy to that knowledge? Does every single member of the Empire just think the Dominion is trying to invade for invasion's sake?

No doubt the Moth Priests are aware.

So by extension, the Elder Council?

Actually, I've not seen any solid evidence for the Thalmor being obsessed with towers.